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PACIFIC JOURNAL OF MATHEMATICS Vol. 160, No. 2, 1993

A CONTINUATION PRINCIPLE FOR PERIODIC SOLUTIONS OF FORCED MOTION EQUATIONS ON MANIFOLDS AND APPLICATIONS TO BIFURCATION THEORY MASSIMO FURI A N D M A R I A PATRIZIA

PERA

We give a continuation principle for forced oscillations of second order differential equations on not necessarily compact diίferentiable manifolds. A topological sufficient condition for an equilibrium point to be a bifurcation point for periodic orbits is a straightforward consequence of such a continuation principle. Known results on open sets of euclidean spaces as well as a recent continuation principle for forced oscillations on compact manifolds with nonzero Euler-Poincare characteristic are also included as particular cases.

0. Introduction. Let M be a smooth (boundaryless) ra-dimensional manifold in W1 and consider on M a time dependent Γ-periodic tangent vector field, i.e. a continuous map / : R x M —> W1 with the property that, for all ( ί , ί ) € R x M , / ( / , ί ) is tangent to M at q and f(t + T, q) = f(t, q). The map / may be interpreted as a (periodic) force acting on a mass point q (of mass 1) constrained on M. A forced (or harmonic) oscillation on M is a Γ-periodic solution of the motion problem associated to the force / . In [FP4], in the attempt to solve the conjecture about the existence of forced oscillations for the spherical pendulum (i.e. for the case M = S2, the two dimensional sphere), we have studied the one-parameter motion problem associated to the force λf, λ > 0. In this context, we say that (λ, x) is a solution (pair) of the problem, if λ > 0 and x: R -> M is a forced oscillation corresponding to λf. Let us denote by X the set of all solution pairs. Since any point q e M is a rest point of the inertial problem (i.e. the motion problem with λ = 0), the constraint M may be regarded as a subset of X by means of the embedding q >-+ (0, q). With this in mind, we say that M is the manifold of trivial solutions of X and, consequently, any element of X\M will be a nontrivial solution (pair). We observe that in the nonflat case one may have nontrivial solutions even when λ = 0. Closed geodesies may be, in fact, Γ-periodic orbits if they have appropriate speed.

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MASSIMO FURI AND MARIA PATRIZIA PERA 1

If we consider the standard C metric structure on the space Cj(M) 1 of all Γ-periodic maps x: R —• M of class C , the main result in [FP4] can be stated as follows: THEOREM 0.1. Assume that the constraint M is compact with nonzero Euler-Poincare characteristic. Then X\M contains an unbounded connected subset whose closure in [0, oc) x Cγ(M) meets M.

In [FP4], an element q of the trivial subset M of X was called a bifurcation point (for the forced motion problem considered above) if any neighborhood of q contains a nontrivial solution, that is, if q is in the closure of X\M. So, as a consequence of the above result, one gets the existence of bifurcation points for a parametrized forced constrained system, provided that the constraint M is compact and χ(M) 9 the Euler-Poincare characteristic of M , is nonzero. We will show that a necessary condition for a point q G M to be a bifurcation point is that the average force / is zero at q i.e. T

Thus, Theorem 0.1 may also be regarded as an extension (or dynamical version) of the classical Poincare-Hopf theorem, which asserts that any tangent vector field on a compact boundaryless manifold M, with χ(M) Φ 0, vanishes somewhere. In fact, observe that a time independent tangent vector field / on M may be regarded as a periodic force (of any period). Theorem 0.1 above was successfully used in [FP5] to give an affirmative answer to the conjecture about the forced spherical pendulum. However, since the problem on whether or not any compact constrained system M with χ(M) Φ 0 has forced oscillations is still unsolved, we think that further investigations about one-parameter forced constrained systems may be of some interest. Our aim here is to extend to the noncompact case some of the results of [FP4], including the one above. The interest of this is mainly related to the fact that open sets in R m are noncompact manifolds. So, the above theorem does not apply to the flat case. Moreover, once the necessary condition for a point q e M to be a bifurcation point is fulfilled, the possibility of dealing with noncompact manifolds will give us the tools to restrict our attention to a neighborhood of q, in order to get sufficient conditions for bifurcation.

PERIODIC SOLUTIONS OF FORCED MOTION EQUATIONS

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In other words, what we do here is the spirit of [FP2] where our effort was devoted to first order differential equations on noncompact manifolds. n One may argue that a motion equation on a manifold M cR is just a special first order differential equation on the tangent bundle T(M)

n

n

= {(q, v) e R x R : q e M, v is tangent to M a t ? } .

However, a bifurcation problem associated to a force of the form λf, where, as above, λ > 0 and / : R x M —> Rn is a Γ-periodic tangent vector field on M, cannot be simply reduced to a problem of the form: z ( t ) = λg(t,z(t))9

ί€R,

z(t)eT(M),

studied in [FP2]. The reason is that the inertial motion problem does not correspond, in the phase space, to the trivial equation z{i) = 0. Actually, as is well known, the motion problem of a mass point q acted on by a force λf has the following form on T(M):

where the term h, which is a (nontrivial) tangent vector field on T(M), is related to the geometry of M and linear only in the flat case. This makes the problem in the non-flat case hard to deal with and cannot be handled with the techniques developed in [FP1] and in [Mar], where, for λ = 0, the problem was linear (with nontrivial kernel). We point out that a very interesting continuation principle for equations of the above form (and not necessarily related to second order equations) is given in [CMZ], where, roughly speaking, the equation is given on an open subset of a euclidean space and the existence of a branch of solution pairs (λ, z) is ensured provided that the Brouwer topological degree of h is (well defined and) nonzero. What seems peculiar to us, and interesting for further investigations, in our situation, is the role of g (or, equivalently, of the force /) which is important for the existence of a bifurcating branch. In fact, as we shall see, it is just the Euler characteristic of the average force / which, when defined and nonzero, ensures the existence of a global bifurcating branch. To see the relation with well-known concepts we recall that the Euler characteristic of a tangent vector field coincides with the Brouwer degree in the flat case and with the Euler-Poincare characteristic of the manifold in the compact boundaryless case.

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MASSIMO FURI AND MARIA PATRIZIA PERA

1. Notation and preliminaries. In this section we recall some definitions and results that will be needed in the sequel. The inner product of two vectors υ and w in Rn will be denoted by (v,w) and \υ\ will stand for the euclidean norm of υ (i.e. \v = (v,v){/2). Let M be an m-dimensional boundaryless smooth manifold in W and, for any q e M, let Tq(M) c W1 and T^M)1- c Rn denote respectively the tangent space and the normal space of M at q . Let T(M) denote the tangent bundle of M, i.e. the 2m-differentiable submanifold = {(q,v)eRnxRn:qeM,

veTq(M)}

of I " x P , containing a natural copy of M, via the embedding Given (